<p>If \(aN = \{ax : x \in \mathbb{N}\}\), then the set \(4N \cap 6N\) is</p>
Step-by-Step Solution
Key Concept: 4N consists of all positive multiples of 4, and 6N consists of all positive multiples of 6. Their intersection contains only numbers divisible by both 4 and 6, which are exactly the multiples of LCM(4,6) = 12.
<p><strong>Step 1:</strong> Write out the sets explicitly.</p><p>4N = {4, 8, 12, 16, 20, 24, 28, 32, 36, ...} = {4x : x ∈ ℕ}</p><p>6N = {6, 12, 18, 24, 30, 36, ...} = {6x : x ∈ ℕ}</p><p><strong>Step 2:</strong> Find common elements (intersection).</p><p>Elements in both sets are: {12, 24, 36, 48, ...}</p><p>These are exactly the multiples of LCM(4, 6).</p><p><strong>Step 3:</strong> Calculate LCM(4, 6).</p><p>4 = 2², 6 = 2 × 3</p><p>LCM(4, 6) = 2² × 3 = 12</p><p><strong>Step 4:</strong> Express the result.</p><p>4N ∩ 6N = {12, 24, 36, 48, ...} = {12n : n ∈ ℕ} = <strong>12N</strong></p><p>∴ Answer: <strong>12N</strong> (Option C)</p>
Correct Answer: C